TMME, vol3, no.2, p.

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An In-depth Investigation of the Divine Ratio
Birch Fett The University of Montana Abstract: The interesting thing about mathematical concepts is that we can trace their development or discoveries throughout history. Most cultures of the ancient world had some form of mathematics, and these basic skills developed into what we now call modern mathematics. The divine ratio is similar in that it was used in many different sections of history. The divine ratio, sometimes called the golden ratio or golden section, has been found in very diverse areas. The mathematical concepts of the golden ration have been found throughout nature, in architecture, music as well as in art. Phi is an astonishing number because it has inspired thinkers in many disciplines, more-so than any other number has in the history of mathematics. This paper investigates how the golden ratio has influenced civilizations throughout history and has intrigued mathematicians and others by its prevalence. Keywords: Egyptian mathematics; Fibonacci; Golden mean; Golden ratio; Greek mathematics; Indian mathematics; mathematical aesthetics; mathematics in nature Introduction Throughout this paper, the terms golden ratio, divine ratio, golden mean, golden section and Phi (φ) are interchangeably used. Wasler, (2001) defines the golden ratio as a line segment that is divided into the ratio of the larger segment being related to the smaller segment exactly as the whole segment is related to the larger segment. The divine ratio is the ratio of the larger segment, AB, of line AC to the smaller segment BC of the line AC.
A B C

This same definition was first given by Euclid of Alexandria around 300 B.C. He defined this proportion and called it “extreme and mean ratio” (Livio, 2002). Let us assume that the total length of line AC is x+1 units and the larger segment AB has a length of x. This would mean that the shorter segment BC would have a length of 1 unit. Now we can set up a proportion of AC/AB = AB/BC.

By cross multiplying it yields x2 – x – 1. Using the quadratic formula, two solutions become apparent (1+√5)/2 and (1-√5)/2, and we only use the positive solution because we are in terms of a length. The positive solution is (1+√5)/2. Phi is the only number that has the unique property that φ*φ'=-1where φ' is the negative solution to the quadratic (1-√5)/2 (Huntley, 1970).

TMME.C. point E is obtained. 2001).1 Barr chose to honor the great sculptor because many of Phidias's sculptors contained the Divine Ratio. Logarithmic spirals are also closely related to Golden Triangles3. The growth patterns in nature approach the golden ration. Phi represents some remarkable relationships between the proportions of patterns of living plants and animals. no. Irrational numbers have been around for sometime. reveal growth patterns that are related to the golden ratio. which is the pole of a logarithmic spiral passing
1
Phidias was an Greek sculptor who lived between 490 and 430 B. but never actually reach the exact proportion (Elam. and the new spiral is extremely close to the proportion of the golden section square larger than the previous. This implies that the shape remains unchanged over time and growth. Contour spirals of shells. triangle ABC has been cut into two isosceles triangles that have golden proportion (the ratio of their areas is φ:1.158
Additional Information on the Golden Ratio In professional mathematical literature. Most historians believe that irrational numbers were discovered in the fifth century B. Each increment in the length of the shell is accompanied by a proportional increase in its radius. vol3. Ram horns and elephant tusks. Continuing this process by bisecting angle C. although they do not lie in a plane.
Logarithmic spirals can be found through-out nature. As a logarithmic spiral grows wider. The symbol (τ) means "the cut" or "the section" in Greek. Mark Barr represented the Golden Ratio as phi (φ). p. the bisector of angle B meets AC at point D and is the golden cut of AC. thus constructing two more golden triangles. The nautilus shell has patterns that are logarithmic spirals2 of the golden section. The golden ratio is a known irrational number. Comment: astonishing is a strange word to use here…how about great? 2 Logarithmic spirals have a unique property. Again point E is the golden cut along line BD. and in some cases come very close to it. Each section is characterized by a spiral. which is the first Greek letter in the name of Phidias. His sculptors included "Athena Parthenos" which is located in Athens and "Zeus" which is located in the temple of Olympia. 4 A gnomon is a portion of a figure which has been added to another figure so that the whole is of the same shape as the smaller figure. such as the chambered nautilus. 3 Golden Triangles are isosceles triangles that exhibit base angles of 72 degrees and an apex angle of 36 degrees. American mathematician gave the golden ratio a new name. From the Pythagoreans and the construction of the pentagram (which has five equal-area golden triangles) it can be seen that the length of the longer side to that of the shorter side is in golden proportion. In the early twentieth century.C. 2001). This process produces a series of gnomons4 that will eventually converge to a limiting point O. With this bisection. The Golden section is aesthetically pleasing in nature. the golden ratio is represented by the Greek letter tau. follow logarithmic spirals. Pythagoreans knew about irrational numbers and believed that the existence of such numbers was due to a cosmic error (Livio.
. the distance between its coils increases and it moves away from its original starting point (pole). A construction of the golden rectangle and logarithmic spiral can be seen below.2. Starting with a Golden Triangle ABC. It turns by equal angles and increases the distance from the pole by equal ratios.

van Iterson showed that the human eye would pick out patterns of winding spirals when successive points were packed tightly together. The familiar spirals that the human eye would pick out consisted of counter clockwise and clockwise patterns of consecutive Fibonacci numbers.
5
. the lengths of these segments form a Fibonacci series. Most sunflowers
Mathematicians. The points were separated by the Golden Angle which measures to 137.625. if the angle used was 360/n where n is an integer. With numerous plants. it is 1:1. vol3. Interestingly.
A
D G F C B E
If we begin with GF and call it the unit length. Harold S. Adler. this mode of curvature is the most prevalent condition of plants lacking or having secondary growth. then: FE = 1φ ED = 1φ + 1 DC = 2φ + 1 CB = 3φ + 2 BA = 4φ + 3 By bisecting the base angles of the successive gnomons. As seen above. which is in proportion of the Golden Ratio. the leaves would be aligned radially along n lines. 1970). 1982). which we have already seen to converge to the Golden Ratio. p. For example. M.D…) of each of the series of the triangles (Huntley. specifically sunflowers. 2001). The seeds of pine cones grow along two intersecting spirals that move in opposite directions. A flexed plant axis is shown to conform to a portion of a logarithmic spiral. The proportion of 8:13 is 1:1. Eight of the spirals move in the clockwise direction and the remaining thirteen move counter clockwise. Sunflowers exhibit the same spiral patterns as seen in pine cones.B. Sunflowers have 21 clockwise spirals and 34 counter clockwise.619 (Elam. grows in the most efficient way5 of sharing horizontal space.TMME. Plants like sunflowers represent this growth pattern (Niklas and O'Rourke.C. Each seed in a pine cone is part of a spiral growth pattern that closely relates to φ. each seed belongs to both spirals.2. Pine cones and sunflowers are closely related to the Golden Ratio. thus leaving large spaces. The geometry of plant axis flexure is the result of orthotropic growth and the stress caused by a vertical weight distribution along the axis. In 1907 the German mathematician G. Using the Golden Angle. no.159
successively and in the same order through the three vertices (…A. Nature. Coxeter and I. the numbers 8 and 13 are consecutive Fibonacci numbers which converge to the Golden Ratio.5 degrees. The proportion of 21:34 is even closer to the Golden Ratio than that of pine cones. showed that buds of roses which were placed in union with spirals generated by the Golden Angle were the most efficient.

In the following sections. Phi can be seen in many places. p. He later published the book Liber Abaci (Book of Abacus). and how to convert between the various currencies in use in the Mediterranean countries. This problem gave the mathematical world the series of Fibonacci numbers7. from the layout of seeds in an apple to Salvador Dali's painting "Sacrament of the Last Supper" (Livio. and that all the rabbits are immortal. 2002). which was a book based on the arithmetic and algebra that he had accumulated in his travels. the ratio of Fn/Fn+1 approaches the golden ratio. This book introduced the Arabic numbering system to Europe and gave Fibonacci everlasting fame as a mathematician. commonly known as Fibonacci6 introduced the world to the rabbit problem.619048
which is an irrational multiple of 360 degrees.160
have a 21:34 ratio. 2001). Fibonacci wrote. Liber Abaci.615385 10 34 1.666667 7 8 1. given that adult rabbits produce a pair of rabbits each month. how to calculate profit on transactions. 1997) 7 Fibonacci Numbers are represented by the recursive relation An=2 = An+1 + An
. vol3. Fibonacci is most remembered for presenting the world with the "rabbit problem" which is located in the third section of Liber Abaci. Most of the problems in Liber Abaci were aimed at merchants and related to the price of goods. Looking at the ratio of successive Fibonacci numbers.D. no. In 1202 A. 6 Fibonacci is a shortened form of Filius Bonaccio (son of Bonaccio). but few have been reported with proportions of 89:55. This book was widely copied and introduced the Hindu-Arabic place-value decimal system and the use of Arabic numerals into Europe. The Golden Ratio can be found in many examples throughout the world. 144:89 and 233:144 (Livio. an in-depth look is taken on the occurrences of Phi in as well as the development of Phi throughout history. As the n increases. The Golden Ratio and Fibonacci Numbers Leonardo de Pisa. The rabbit problem asked to find the number of rabbits after n months. offspring take one month to reach reproductive maturity.D.6 8 13 1.TMME. (Dunlap.2. born around 1175 A.5 6 5 1. ensures that the do not line up in a specific radial direction and this leaves no space unfilled. Fibonaci was taught the Arabic system of numbers in the 13th century.625 9 21 1.. The values (n=1…10) can be seen in the table below: n F(n) F(n)/F(n-1) 1 0 2 1 3 1 1 4 2 2 5 3 1. an interesting value appears.

A Scottish mathematician. A geometric sequence can be constructed on the basis of the breeding rabbits. Interestingly. Plato created a chemistry that is similar to modern day chemistry9 (Livio. when water is heated by fire. the octahedron with eight triangular faces. and dodecahedron could be constructed out of two types of right angled triangles.8 He divided the heavens into four basic elements. can be written as AbAAbAbAAbAAbAbAAbAbA… The sequence of A’s and b’s may be extended indefinitely in a unique way because the rule for generating the next character is well defined. balancing the number of faces involved (in the Platonic solids that represent these elements) we get 20=2*8+4.) prophesied the significance even before Euclid described it in Elements. limn→φ A/b = φ The Golden Ratio in Ancient Greece The Golden Ratio can be found throughout nature. each of the solids can be circumscribed by a sphere with all of its vertices lying of the sphere. The arrangement of adults (A) and their offspring (b). were based on equilateral triangles. {water}→2{air}+{fire}. which will be discussed below. Plato saw the world in terms of perfect geometric proportions and symmetry. Plato (428-347 B. For example. noticed that consecutive terms of the solution to the rabbit problem converged to the Golden Ratio (Johnson. 2002). The tetrahedron consisted of four triangular faces.
8
. octahedron. earth. the dodecahedron with twelve pentagonal faces and the icosahedron with twenty triangular faces (Livio. The central idea is that particles in the universe and their interactions can be described by a mathematical' theory that possesses certain symmetries. In order
The Platonic Solids Plato used consisted of five shapes. In Platonic chemistry. vol3. it produces two particles of vapor (air) and one particle of fire. The five Platonic solids are the only existing solids in which all of the faces are identical and equilateral and each vertex is convex. p. Robert Simson (1687-1768). octahedron and the icosahedron. tetrahedron. 1999). The remaining two. the isosceles 45-90-45 and the 30-60-90 triangle. Each face of the regular polyhedron is a regular polygon with n edges. tetrahedron for fire. but it can also be found in the history of the heavens. air. a cube for earth. Each of these elements was assigned a Platonic Solid. For example. 2003). Using this foundation. no.TMME. It is known that the values of n are {n: 3≤n<∞} with n being related to the interior angle α. cube. where m≥3. let adult rabbits be represented by ‘A’ and their offspring represented by ‘b’.161
The convergence of Fibonacci numbers to the Golden Ratio can be seen in the “rabbit problem”. Plato explained that his chemical reactions could be described using these properties. octahedron for air and an icosahedron for water. He noted that the faces of the tetrahedron. cube and dodecahedron were made from the square and regular pentagram.C. 10 In general a regular n-gon has n edges and interior angles given by the equation α=[1-(2/n)]*180.2. 9 Plato's theory was much more than a symbolic association. water. 1997). The first three. in the early 1700's made the connection between the Golden Ratio and the rabbit problem. the cube with six square faces. and fire. His ideas were based on Platonic Solids.10 Each vertex of the three dimensional polygon is defined by the intersection of a number of faces. m. The ratio of adults to offspring rabbits in the limit of an infinite sequence is equal to the Golden Ratio (Dunlap.

. 12 The table above lists the characteristics of the five Platonic Solids.TMME. faces. There are many different accounts of the Mathematician’s life and death. Many authors researching ancient Greek mathematics are unsure if the works of Plato were influenced by Pythagoras and the Pythagoreans. These reasons are if m=2 then an edge is formed. vol3. but what is known for sure is that he was responsible for mathematics. He studied Egyptian. p. no. The number two was considered the first female number and the number of opinion and division. 2002).C. Pythagoras13 was born around 570 B. then the vertex is merely a point on a plane and if mα>360 degrees then the faces overlap. 13 Pythagoras emigrated to Croton in southern Italy sometime between 530 and 510. However. 1997). Plato and his followers may have created and used Platonic Solids in the foundations of the universe based on sheer beauty. but both of these prove too applied for him. The quantities e.1997). Pythagoras and the Pythagoreans are best known for their role in the development of mathematics and for the application of mathematics to the concept of order (Livio. 11 There are only five combinations of integers that satisfy these equations and they correspond with the five Platonic Solids and are listed below12 (Dunlap . and philosophy of life and religion. The quantities n and m are the number of edges per face and the number of faces per vertex. solid dodecahedron icosahedron surface area 15φ/(3-φ) 5√3 volume 5φ3/(6-2φ) 5φ5/6
Plato and his foundations using Platonic Solids for the heavens may suggest that the Golden Ratio may have been known in ancient Greece. And if mα=360 degrees. on the island of Samos. and v are the total number of edges. it is easy to see the important role the Golden Ratio play in their dimensions (Dunlap.162
for a convex vertex to be formed. and vertices for the respective solid. the number two was expressed by the line
11
We have to place certain restrictions on the values of m. mα<360 degrees. The Pythagoreans assigned special properties to odd and even numbers as well as individual numbers. Geometrically. not a vertex. and Babylonian mathematics. solid tetrahedron cube (hexahedron) octahedron dodecahedron icosahedron n 3 4 3 5 3 m 3 3 4 3 5 e 6 12 12 30 30 f 4 6 8 12 20 v 4 8 6 20 12
The Golden Ratio is of relevance to the geometry of figures with fivefold symmetry. the generator of all dimensions. The number one was considered the generator of all other numbers and geometrically. If either one of these Platonic Solids are constructed with an edge length of one unit. The dodecahedron and the icosahedron are of particular interest.2. the full mathematical properties of Platonic Solids may not have been known in antiquity. f.

The main reason five is important to this discussion is because the Pythagoreans used the pentagram14 as a symbol of their brotherhood (Livio. sliding the straight edge through point B until one of the points falls on the arc of A. Justice and order was expressed in the number four. Six is a perfect number because it is the sum of all the smaller numbers that divide into it. The fifth vertex (E) can be found by the requirement that on line EGB. not in the same plane. find points G and D. C and F. The construction of the pentagon. namely.
D
E F
P a G A a B
C
D
E
d
c
C
f a
b
Q
A
B
14
The pentagram is closely related to the regular pentagon. 6 = 1+2+3 28 = 1+2+4+7+14 496 = 1+2+4+8+16+31 The number five deserves its own explanation.TMME.163
which has one dimension.
. 2002). Given a line AB. The number three is considered by the Pythagoreans to be the first male number and the number of harmony because it combines the unity number (one) and the division number (two). use the compass to draw arcs of radius a about points A and B. no. This process can be continued to infinity. where the area of the triangle has two dimensions. There are only two possible positions for these points. Using this construction of a pentagon. Next construct the perpendicular bisector PQ of line AB. On the surface of the Earth. The number six is the first perfect number and considered the number of creation. using a compass and marked straight edge. p. Five represents the union of the first female number and the first male number. 2002).2. and every segment is smaller that its predecessor by a factor that is precisely equal to the Golden Ratio. The diagonals of this pentagon form a smaller pentagram. If one is to connect all the vertices of the pentagon by diagonals. form a tetrahedron. EG equals a. It is the number of creation because it is the product of the first female number (two) and the first male number (three). four directions provide orientation for humans to identify their coordinates in space. Using the straight edge plot two points that are a units apart and slide the straight edge so that it passes through point A. This union suggests that five is the number of love and marriage. 1987). leads to a pentagram. Using the same directions. one can connect the vertices and build a pentagram (Herz-Fischler. The first three perfect numbers are listed below (Livio. until one of the points falls on the arc of B. vol3. Four points. a pentagram is constructed. The geometric expression of the number three was a triangle.

dn-1. For pentagonal side and diagonal numbers. Ancient civilizations did not necessarily have the same numbering systems of modern times. vol3. This suggests that some things that work in modern numbering systems do not work in ancient systems (Rossi and Tout. The diagonals of a pentagon cut each other in the Golden Ratio and the larger of the two segments is equal to the side of the pentagon.2. suggests that the Pythagoreans were familiar with the Golden Number. p. due to inconclusive historical data (Herz-Fischler.
15
Side and diagonal numbers of squares start off with the number one as the first number in the sequence. sn.164
The pentagram is important to the discussion of the Golden Ratio because of its unique properties. This may prove useless because it can produce an infinite chain of similar links. Heller (1958). 1987). Ancient Egypt. a civilization with profound mathematical accomplishments and astonishing monuments is under investigation for uses of the golden mean. sn-1 and dn-1. One theory. The Pythagoreans choosing the pentagram as a symbol for brotherhood. starting with one will lead to the degenerate case. suggests that the Pythagoreans used the pentagon to discover incommensurability and the division in extreme and mean ratio.
. 1987). and the given properties of the pentagram. Many interpretations of the golden mean use the properties of different geometrical figures. Heller believes that the Pythagoreans discovered incommensurability through the observations of a series of pentagons when drawing diagonals. Using15 s1 = 2 and d1 = 3. The new diagonal dn is the sum of the side and the diagonal. but many historians are still under debate about this particular topic. Math historians do need to focus on the ancient monuments and the mathematics of the respective time period. 13/8… which we have already seen to be successive Fibonacci numbers (Herz-Fischler. 2002).TMME. Thus we have to start with the two as the first number in the sequence. no. of the next largest pentagon. 5/3. dn = dn-1 + sn-1. leads to the sequence of dn:sn ratios of 3/2. of the previous pentagon. 8/5. The Golden Mean in Ancient Egypt Modern mathematicians have been trying to decide what civilizations used and understood the golden mean.
dn-1 sn=dn-1 dn=dn-1+sn-1
sn-1
The diagonal. becomes the side. A formal proof of this can be found in The Golden Ratio: The Story of Phi the World's most Astonishing Number. With this information it is easy to see the recurrence relationships sn = dn-1.

p. 1997). Using the Pythagorean Theorem.
h
s
b
In the above figure. has been measured and many different dimensions are present. 1992). The majority of the dimensions are within one percent of 755.. We have already seen that the ratio of corresponding Fibonacci numbers converges to the Golden Ratio (Rossi and Tout.165
One theory about the use of the Golden Mean in ancient Egypt is that Egyptian architects designed the pyramids in a geometric way. This gives us a ratio of the slant height of the pyramid to half the length of the base as 612. Using 755.C.4 feet for the height. Egyptians could have represented this number in five different ways: 1.90=1.TMME.4 feet as the height. which has dimensions derived for the Golden Ratio. Some theories claim that the Great Pyramid of Cheops was designed so that the ratio of the slant height of the pyramid to half the length of the base would be the divine proportion (Markowsky. h2 + b2 = s2. 1/2 + 1/10 + 1/56 + 1/840 2. 1992). was it possible for ancient Egyptians to construct a convergence of the Fibonacci numbers? Ancient Egyptians represented ratios as a sum of unit fractions.43 degrees. Take the ratio 13/21 for example. we can see that b=377. As ratios continued to grow. The question that needs to be answered is.
.90 feet. 2002).01. h represents the height.79 feet for the length of the base and 481. rectangles and triangles. The Great Pyramid of Cheops. Egyptian pyramids were based on geometrical processes of squares. many different representations become available.2.62 which is very close to the Golden Mean (Markowsky. For example the fraction 3/5 would be represented as 1/2 + 1/10. This is very close to the apex angle of the Golden Rhombus17 (63. The theory continues to suggest that Egyptian architects gave their designs dimensions based on the corresponding numbers of the Fibonacci series. we can find that s=612. 1/2 + 1/10 + 1/57 + 1/665 3.79 feet as the length of the base and 481. built before 2500 B. The Golden Rhombus is a two dimensional figure that has perpendicular diagonals which have a ratio of 1:φ. 1/2 + 1/10 + 1/65 + 1/273
16 13
The 8:5 triangle was an isosceles triangle in which the base was eight units and the height was five units.435 degrees). Another interesting feature of the Great Pyramid is that it has an apex angle of 63. and s represents the slant height of the Great Pyramid of Cheops. Of extreme importance was the process of the 8:5 triangles.01/377.16 Egyptians used these triangles because the ratio of 8/5 was a good approximation of the Golden Mean. The difference between the apex angle of the Great Pyramid and a Golden Rhombus is a mere 22 centimeters in the edge of the length of the pyramid base (Dunlap. vol3. 1/2 + 1/10 + 1/60 + 1/420 4. b represents half the base. 1/2 + 1/10 + 1/63 + 1/315 5. no.

1/2 = 1/2 3/5 = 1/2 + 1/10 8/13 = 1/2 + 1/10 + 1/65 21/34 = 1/2 + 1/10 + 1/65 + 1/442 55/89 = 1/2 + 1/10 + 1/65 + 1/442 + 1/3026 144/233 = 1/2 + 1/10 + 1/65 + 1/442 + 1/3026 + 1/20737 The convergence above suggests that is was possible for ancient Egyptian scribes to evaluate the Golden Ratio. With this. However. The circumference of the circle is divided into 360 degrees and then the radius of the circle is divided into 60 parts. Egyptian math is considered an applied math. 2002). A proof of this statement is provided by Gupta (1976) and is provided below. multiply the diameter by 70534/12000 (Amma. it seems unlikely that ancient Egyptians were aware of the Fibonacci numbers. This is exactly the relationship sine (18) = a10/2. The sum of the ratios of the first few Fibonacci numbers converging to φ can be seen below (Rossi and Tout. sine (30) = a6/2 = r/2 = 30. And sine (18) = a10/2 and sine (36) = a5/2. no. did not recognize the golden ratio and it was a mere coincidence that the architecture of the pyramids is based on 8:5 triangles (Rossi and Tout. The Indian sine function can be defined as satisfying the relationship Sine (θ) = ½* chord (2θ). Adding to the previous sum of ratios a unit fraction whose denominator is given by the multiplication of the two previous denominators (in the ratio of Fibonacci numbers) yields the next value in the sum converging to the Golden Ratio. Only applications of Egyptian mathematics exist. The Golden Ratio in Ancient India The division in extreme and mean ratio appears in mathematical texts from India in connection with trigonometric functions. Bhaskara. again without reason. This suggests that the Egyptians. vol3.TMME. 2002).
.166
Egyptian scribes could have found a convergence of φ with their system of representing fractions. The Indian sine function is not the same as our modern day sine function. tells to find the side of the pentagon inscribed in a circle. although capable.2. no records have been found on the theory behind their mathematics.
chd (θ) θ arc
sine (θ)
chd (2θ)
Bhaskara II (1114-1185) states without proof that Sine (18) = (R(5r2) – r)/4. 1979). p.

no. The Golden Ratio has influenced classical Greek architecture. Zeus' first wife. p. Gupta (1976) continues and completes the construction by: Think of Y and C as given points and draw the arc OTX of radius r/2. the artist or workmen unconsciously employ golden proportions.167
In a circle of radius r = OX = OY. war. Many assertions claiming that the Golden Section was used in art are associated with the aesthetics of the proportion. while golden proportions are pleasing to both hand an eye (Ackermann. which has also shown to have Golden proportions. the arts. Thus. Several decades after the Brotherhood of the Pythagoreans faded. Inside the Parthenon stands a forty-foot-tall statue of the Greek Goddess Athena18. Although most humans cannot decipher between a rectangle with a ratio of 1. Irregular inequality and capricious division is aesthetically disagreeable. Her father was Zeus and her mother was Metis.2.6 and a rectangle with ratio of 1. The Golden Section follows upon the basis of symmetry everywhere and the forms which are based upon the golden proportion are widely distributed.TMME. Draw a semicircle OX about the midpoint C of OX and draw the arc MD about Y.7. How does this construction tie in with the discussion on Ancient Indians knowing the Golden Ratio? Concentrate on the triangle YOC and arcs DT and OT. it suggests that humans do prefer rectangles in the range close to the Golden Rectangle (Markowsky. who is the first artist known to use the Golden
18
Athena is the Greek goddess of wisdom. most people would choose rectangles with a close approximation of the Golden Rectangle. When given an opportunity to choose the most visually pleasing rectangle. Assume that the tow arcs meet at the single point T on line YC. vol3. Draw arc MTD of radius YT.
. 1895). there is no equal symmetry. justice and skill. With a close examination. the Golden Ratio continued to influence many artists and artisans. the circles are tangent at the point T on the line YTC connecting the centers. notably the Parthenon in Athens. 1992). YM = YD is the greater segment when OY is divided in extreme and mean ratio (Gupta. industry. it can be seen that OY is divided in extreme and mean ratio at D. Evidence of the Golden Ratio in the Arts Countless illustrations of the proportions of the Golden Section are found in the works of humans. In other words. 1976). Both the temple and the stature were designed by Phidias. let the arc YM = 36 degrees. When speaking about the products of art and architecture.
36 degrees
M T X
C r/2 O
chd 36 degrees
Y
D
The above proof and construction are considered incomplete because they do not explain why the arcs meet at point T. Then Sine (18) = YM/2 = YT/2 = YC/2 –TC/2 = (R(r2 + (r/2)2) – r/2)/2 which is equivalent to Sine (18) = (R(5r2) – r)/4.

If rabatment is applied to a Golden Rectangle." in Athens was built in the fifth century B. who is commonly known as Vitruvius. Renaissance artists often used diagonals and other interior lines of rectangles to divide rectangular space proportionally. its builders had no knowledge of the Golden Ratio. the Parthenon fit precisely into a Golden Rectangle."
. p.168
Ratio in his work. and may get some of its beauty from the regular rhythms introduced by the repetition of the same column (Livio. know as "the Virgin's place in Greek. both vertically and horizontally. 20 On September 26. the diagonals of the halves allow division into quarters. Regardless whether or not the Parthenon's architecture was built accordingly to the Golden Ratio. 2001). The Parthenon is a sacred temple to the cult of Athena Parthenos. Continuing.2.C. Venetian artillery directly hit the Parthenon.
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The Parthenon. advised that "the architecture of temples should be based on the likeness of the perfectly proportioned human body where a harmony exists among all parts" (Elam. The Parthenon can be inscribed by a Golden Rectangle (Elam. Ancient Greek scholar and architect Marcus Vitruvius Pollio. A construction of division by diagonals is provided on the left and a construction by rabatment is provided on the right. and is one of the world's most famous structures. the symbol for the Golden Ratio is the first Greek letter phi. General Konigsmary said "How it dismayed His Excellency to destroy the beautiful temple which had existed for over three thousand years. the diagonals of the two overlapping squares cut the diagonals of the rectangle in golden proportion (Brinkworth and Scott. no.Markowsky. 2001). 2001). 2001).TMME. This concept was the same as the module of human proportions and became an important architectural idea. Another claim is that the height of the structure (from the top of the tympanum to the bottom of the pedestal) is divided into the Golden Ratio (Livio. He believes that even though the Parthenon incorporates many geometric balances. For example the main diagonals of a rectangle allow for division of the rectangle into halves.Vitruvius is credited with introducing the concept of a module to the architectural world. (1992) has a contrasting view of the Parthenon. it is still an amazing structure. This implies that if the author is a Golden Ratio enthusiast they could choose which ever numbers give them the best approximation of φ. Rabatment is where the shorter sides of the picture rectangle are rotated onto the longer. the dimensions of the Parthenon vary because the authors are measuring between different points. The rotation creates vertical division and overlapping squares. The Parthenon 19 in Athens is an example of this proportioning. vol3. Another tactic used by Renaissance artists to construct they work was called rabatment. which also happens to be the first letter in Phidias's name (Johnson. Depending on what sources are used. As said above. 2001). 1687. 1999). When the triangular pediment was still intact 20 .

" Both the painting as a whole and the central figures in the painting can be inscribed by Golden Rectangles. the dimensions are in proportion 1. Jerome” (Markowski. Both Madonna and Child are surrounded by angles. the three Madonnas were painted centuries before the publication of "The Divine Ratio" which brought the proportion into common knowledge (Livio. The drawing of "a head of an old man. it is very possible to find some ratio that approximates the Golden Ratio. was done before da Vinci had any contact with Pacioli or his book "The Divine Ratio." the two version of "Madonna on the Rocks. 2001). Sienese artist Duccio di Buoninsegna's (1255-1319) "Madonna Rucellai" and Florentine painter Cenni de Pepo's (1240-1302) "Santa Trinita Madonna" can be inscribed by Golden Rectangles.58. both close estimates of φ (Livio. In the first version. Leonardo was not introduced to Pacioli’s book until thirteen years after the completion of “St. 2001). p. produced between 1483 and 1486. The first version. Italian painter and architect Giotto di Bondone (1267-1337) painted the "Ognissanti Madonna"21 which is also known as "Madonna in Glory. 1992). The left side of the Golden Rectangle is tangent to a small fold of fabric and does not touch the body at all. Leonardo da Vinci's "St." the drawing of "a head of an old man. When overlaid with a Golden Rectangle. Interestingly. The two versions of "Madonna on the Rocks" have an interesting history. Human body proportions and facial features share similar mathematically proportioned relationships as other living organisms. Leonardo da Vinci’s “head of an old man”. With respect to the time period.2. The placement of facial features yields the classic proportions used by both the Romans and Greeks. His right arm also extends beyond the rectangle's side. Similarly. Jerome" has similar uncertainty. no. Jerome. rather they were driven by the unconscious aesthetic properties of the Golden Ratio. Marcus Vitruvius Pollio described the height of a well proportioned man is equal to the length of his outstretched arms.TMME.
. Again." and the most famous of all. Five of his works have been speculated to host Golden Ratio properties: The unfinished canvas of "St. Leonardo da Vinci inevitable comes into the discussion of the Divine Ratio and art. The rectangles are very roughly drawn and do not have square corners (Markowski. This painting features an enthroned Virgin with a child on her lap.169
In the thirteenth century three artists' work contain close proportions to the Golden Rectangle. the human body is divided into two parts at the naval. Both of these paintings are in the same room as the "Madonna in Glory"."22 completed in 1490. is the closest demonstration that da Vinci used Golden Rectangles to determine dimensions in his paintings (Livio. It is speculated that these three artists did not include the Golden Section into their paintings. could have been influenced by Pacioli's book. 2001). Jerome's body and his head are missed completely. This suggests that depending on where one measures from. Some of these rectangles approximate Golden Rectangles but it is difficult to be absolutely sure." The second version. the left side of St. vol3. The body and outstretched arms can be inscribed in a square. 22 The drawing of "a head of an old man" is currently in the Galleria dell' Accademia in Venice. both versions are very close to the Divine Ratio. These parts are
21
Bondone's painting "Madonna in Glory" is currently in the Uffizi Gallery in Florence. the "Mona Lisa"(Livio. while the hands and feet are inscribed in a circle. which was completed around 1506. 2001). 1992). With this system.64 and in the second version's dimensions are in proportion 1. is suggested to be a self-portrait which is overlaid with a square that is divided into rectangles.

A table of some of Mozart’s movements is listed below. Many listeners. Visually pleasing art is not the only form of art where the Golden Ratio can be found. 279. 25 Mozart wrote 19 all together. However close the approximations are. When reviewing the first movement of the first sonata. The artist. It is divided so that the Development and Recapitulation section has a length of 62. In his early composing years. K. by putting two-measure melodic fragments together in a specific order. he took up the problem of composing minuets ‘mechanically’. including people who are only casually acquainted with the music of Mozart (1756-1791). Almost all of his sonatas were composed of two movements: 1) the Exposition in which the musical theme in introduced and the Development and Recapitulation in which the theme is developed and revisited (Newman. This is true for the second sonata which has total length of 74 and is divided in 28 and 46. no matter how accurate. complete with proofs of how certain data can be transformed to exhibit Golden Ratio characteristics.
Fischler (1981) gives a detailed description. vol3. 24 When Mozart was learning arithmetic. 2001). The Golden Ratio is related to many forms of music. only provide reasonable estimates of the Golden Ratio (Fischler. His sister recalls that he once covered the walls of the staircase and of all the rooms in their house with figures. they could have been created with beauty in mind and with no intention to match the Golden Ratio23. It should be noted that measurements. A visual representation of Mozart’s sonata-form movement can be seen below. then moved to the neighbors house as well (King. can pick up on the manifested form and balance the composer used when writing his music (Putz. Classical statues from the fifth century such as Doryphoros the spear bearer and Zeus have the proportions suggested above (Elam. By the age of nineteen. painter or sculptor may or may not have been trying to conform to the proportion of the Golden Ratio. The art described above deal with proportions of measurements. is 100 measures in length. Mozart worked with mathematical figures throughout his life. 1976).
The first movement of the first sonata. he gave himself entirely to it. Mozart had composed his first sonata for piano 25 .
23
.TMME. p. it can be seen that 100 cannot be divided any closer (using natural numbers) to the Golden Ratio than 38 and 62. 1995). 1981). It should be noted that the lengths of the movements are natural numbers because they measure counts. 1963).170
represented in the proportion of the Golden Rectangle. Music and mathematics have been entwined since antiquity and it is not surprising that one accompanies the other24.2. no.

III 280.59614205x + 2. I 279. Proof: Let x = a/b.TMME. Theorem: │{b/(a+b)} . is consistent with using the Golden Ratio in their works. III
a 38 28 56 56 24 77 40 46 15 39
b 62 46 102 88 36 113 69 60 18 63
a+b 100 74 158 144 60 190 109 106 33 102
To evaluate the consistency of the ten proportions listed above. I 281. then both a/b and b/(a+b) should be near phi. a scatter plot of b against a+b can be used. I 280. The statistical analysis of the data and the graphs below show that the data is linear and the points scarcely differ from the line y = φx. I 282.171
Piece and Movement 279.994 which confirms an extremely high degree of linearity. II 280. Fischer (1981) provides a theorem and the following proof that b/(a+b) is always closer to φ than a/b is. no. II 282. III 281. Mozart in this case. If a composer. II 279.
Degree of Consistency 200 180 160 140 120 100 80 60 40 20 0 0 20 40 60 a+b 80 100 120 b
Degree of Consistency 200 180 160 140 120 100 80 60 40 20 0 0 20 40 60 a+b 80 100 120 b
If a movement is divided into the Golden Section. The graph on the left represents the degree of consistency by plotting the value of b with the values of a+b. The statistical analysis for the data shows an r2 value of . and the line y = φx (black line) overlaid on the plot of the data.733467326). the data should be linear and fall near the line y = φx. 1995).φ│ ≤ │(x) – φ │
. The graph on the right shows the linear regression of the data (represented by the yellow line and the equation y = 1.φ│ ≤ │(a/b) – φ │where 0 ≤ a ≤ b. │{1/(x)} . p. vol3. This is of impressive evidence that Mozart did partition sonata movements near the Golden Section (Putz.2. Then show that.

A single DNA molecule measures 34 angstroms long by 21 angstroms wide for a full cycle of its double helix spiral.
. The ideal face can be measured in symmetrical proportions. The double-stranded helix DNA molecule has two grooves in its spiral. It has been suggested that this represents a geometrically encoded instructional pattern in the brain that guides humans to recognize beauty.” Facial harmony can be activated through symmetry. charm or coloring. The major groove measures 21 angstroms and the minor groove measure 13 angstroms. A simple calculation will show that φ is a fixed point of f.φ│ ≥ │(x) – φ │ with equality when x = φ. A modern definition of beauty is “excelling in grace or form. Both 34 and 21 are Fibonacci numbers which converge to the Golden Ratio. that is. This theorem says that the ratio of consecutive terms of any Fibonacci-like sequence (f1 = a. Let f(x) = 1/(x+1). vol3. The mouth and nose can each be placed at Golden Sections of the distance between the eyes and the bottom of the chin. This is exactly how some modern plastic surgeons are creating beauty. Another unique way that DNA is related to the Golden Number can be seen in a cross-sectional view of a DNA strand. p.618. which turns out to be a decagon. Such symmetry exists when one side of the face is a mirror image of the other.1].1). no. The Golden Proportion can be found throughout a beautiful human face. The golden properties of the decagon are discussed above. 2003). Now f’(x) = -1/(x+1)2 satisfies ¼ < │f’(x) │< 1 For x Є (0. With this information it is possible to construct a human face with dimensions exhibiting the Golden Ratio.2. fn+2 = fn + fn+1 with a and b not both zero) converges to φ. The Golden Ratio can also be found in human DNA structure26 and has been found to be the only mathematical configuration that can duplicate itself ad infinitum without variance. for all x Є [0. The Golden Decagon Mask is completed when
26
DNA molecules are based on the Golden Ratio. The human head forms a Golden Rectangle with the eyes at the midpoint. Dr.172
for all x Є [0.1] there is a z Є (0. for all x Є [0. which is a two-dimensional visual perception of the face that has triangles with sides with ratios of 1:1. It should be noted that attractive faces are relatively symmetrical but not all symmetrical faces are considered beautiful (Adamson & Galli.1]. │{1/(x+1)} . So. f2 = b.1) such that: │f(x) – f(φ)│ = │f’(z) ││ x – φ │. By the Mean Value Theorem. both are Fibonacci numbers. qualities which delight the eye and call forth that admiration of the human face in figure or other objects. Modern Implications of the Golden Ratio and Beauty Beauty has been defined in many different ways since antiquity. Stephen Marquardt created a Golden Decagon Mask.TMME. that f(φ) = φ. again.

TMME. however we cannot deny the principles that accompany it. and architecture. no. a vertex radial and an intersect of two vertex radials. Egypt 500 B.C. Greece 164 A. Rome 1794 A.D. music. throughout history.2. vol3. Plastic surgeons may construct beautiful faces today to fit into a Golden Decagon.” Concluding Thoughts Phi could be the world's most astonishing number. Below are some examples of how the Golden Ratio is perceived throughout history and through different cultures. beauty will always be defined in more ways than one. or two intersects of vertex radials in common with the primary Golden Decagon matrix. Many conflicting theories exist about the origins of phi (φ). But it does make you wonder if “beauty is in the phi of the beholder. These secondary Golden Decagon Matrices form the various features of the face (Marquardt. It can be found in nature.
Asian
Black
Caucasian
Regardless of how the human face seems to fit into a unique geometric figure. The future may lead to a new definition of beauty based on other information than the golden ratio.C.173
forty-two secondary Golden Decagon matrices 27 are mathematically and geometrically positioned in the primary framework.
1350 B. in art.D. but this may not always be the case. 2002).
. p. Whether it is the mathematical
27
The secondary Golden Decagon matrices are constructed exactly the same way as the primary Golden Decagon only smaller. The secondary matrices are geometrically locked on to the primary matrix by having at least two vertex radials.

Herz-Fischler. & Galli. NY: Dover Publications. 406-410. 24(1). P. 1-10. S. P. Livio. 256-262. R. Huntley. (1999). 4(4). 31(4). New York: Dover Publications. M. Mozart in retrospect. (1981). M. 11 (1) (1976). the world’s most astonishing number. Fiber meets Fibonacci. The American Mathematical Monthly. Fibonacci Quarterly.174
relationships that seem to form around the number or the sheer aesthetics of the proportion. H. (2001). Mineola. (2003). 57(3). (1976) Sine of eighteen degrees in India up to the eighteenth century.TMME. R. Plastic Surgical Nursing. Johnson. Acknowledgement The author wishes to thank Professor Bharath Sriraman for his valuable input on various drafts of the paper. The golden section. The last supper at Milan. Astronomy. Fischer. H. K. (1976). Australian Mathematics Teacher. Delhi: Motilal Banarsidass. 260264. Dunlap. 52-58. Singapore: World Scientific.2. New York: Broadway Books. Adamson. Inc. Searching for the golden ratio. we must be aware that φ is all around us and rightly called the Divine Ratio. Modern concepts of beauty. 2-5. The divine proportion: A study in mathematical beauty. R. R. Geometry in ancient and medieval India. 3236. New York: Princeton Architectural Press. E. & Scott. (1997). Indian Journal of History of Science. vol3. Brinkworth. Livio. References Ackermann. A mathematical history of the golden number. King. Gupta. CT: Greenwood Press. E. P. (1979). p.
. The golden ratio and Fibonacci numbers. no. The golden ratio: The story of Phi. A. How to find the “golden number” without really trying. (2003). 2(9/10). 19. (2002). (2001). Westport. (1970). (1895). (1987). S. Geometry of design: Studies in proportion and composition. Mathematics Teaching in the Middle School. A. Amma. Elam.